Question : Two circles with centres A and B touch each other externally, PQ is a direct common tangent which touches the circle at P and Q. If the radii of the circles are 9 cm and 4 cm, respectively, then the length of PQ (in cm) is equal to:
Option 1: 5
Option 2: 13
Option 3: 6.5
Option 4: 12
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Correct Answer: 12
Solution : Let $r{_1}$ and $r{_2}$ be the radii of the two circles, and let $d$ be the distance between their centres. In this case: $d=r{_1}+r{_2}$ Given that the radii are 18 cm and 8 cm, we have: $d=9+4=13\ \mathrm{cm}$ Now, the direct common tangent is the line segment that joins the points of contact of the two circles. This forms a right-angled triangle with the line segment connecting the centres. The length of the direct common tangent can be found using the Pythagorean theorem. Let $t$ be the length of the direct common tangent. Then: $t^2=d^2−(r{_1}−r{_2})^2$ $⇒t^2=13^2−(9−4)^2$ $⇒t^2=169−5^2$ $⇒t^2=169−25$ $⇒t^2=144$ $\therefore t=12\ \mathrm{cm}$ Hence, the correct answer is 12.
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Question : Two circles of radii 9 cm and 4 cm, respectively, touch each other externally at point $A. PQ$ is the direct common tangent of those two circles of centres $O_1$ and $O_2$, respectively. The length of $PQ$ is equal to:
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